The negation of the statement pattern $\sim p \vee (q \rightarrow \sim r)$ is

What does the following figure represent?

If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$,then a value of $\cos \left(\theta-\frac{\pi}{4}\right)$ among the following is

Assume each reaction is carried out in an open container. For which reaction will $\Delta H = \Delta E$?

Consider the following statements about manganate and permanganate ions. Identify the correct statements:
$(A)$ The geometry of both manganate and permanganate ions is tetrahedral.
$(B)$ The oxidation states of $Mn$ in manganate and permanganate are $+7$ and $+6$, respectively.
$(C)$ Oxidation of $Mn(II)$ salt by peroxodisulphate gives manganate ion as the final product.
$(D)$ Manganate ion is paramagnetic and permanganate ion is diamagnetic.
$(E)$ Acidified permanganate ion reduces oxalate, nitrite and iodide ions.
Choose the correct answer from the options given below:

If $(1, a)$ and $(b, 2)$ are conjugate points with respect to the circle $x^2+y^2=25$,then $4a+2b$ is equal to